Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem

The Kochen-Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incom-patibility, entanglement and Bell non -locality are not necessary to demon-strate KS-contextuality. However, here we find that for multiqubit systems, en-tanglement and non-locality are both es-sential to proofs of the Kochen-Specker theorem. Firstly, we show that unen-tangled measurements (a strict super-set of local measurements) can never yield a logical (state-independent) proof of the KS theorem for multiqubit systems. In particular, unentan-gled but nonlocal measurements- whose eigenstates exhibit nonlocal-ity without entanglement-are insuf-ficient for such proofs. This also im-plies that proving Gleason's theorem on a multiqubit system necessarily re-quires entangled projections, as shown by Wallach [Contemp Math, 305: 291-298 (2002)]. Secondly, we show that a multiqubit state admits a statistical (state-dependent) proof of the KS the-orem if and only if it can violate a Bell inequality with projective measure-ments. We also establish the relation-ship between entanglement and the the-orems of Kochen-Specker and Gleason more generally in multiqudit systems by constructing new examples of KS sets. Finally, we discuss how our results shed new light on the role of multiqubit contextuality as a resource within the paradigm of quantum computation with state injection.

Automated summary

The Kochen-Specker theorem shows the nonclassicality of single quantum systems, while Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. However, for multiqubit systems, both entanglement and non-locality are essential to proving the Kochen-Specker theorem.